Christoffel symbols

Provides the Christoffel symbols of the first kind \(\Gamma_{ijk}\) with respect to the Levi Civita connection for a given metric tensor.

Usage

christoffel(g)

Arguments

g

A covariant metric tensor, a "metric_field" object. See metric_field() to create a new metric tensor, or use predefined metrics, e.g. g_eucl_cart().

Value

Returns the Christoffel symbols of the first kind \(\Gamma_{ijk}\) as rank 3 array().

Details

The Christoffel symbols are a rank 3 array of numbers.

See also

Wikipedia: Christoffel symbols

Other geometric tensors: ricci(), ricci_sc(), riemann()

Examples

christoffel(g_eucl_sph(3))
#> , , 1
#> 
#>      [,1] [,2]            [,3]                        
#> [1,] "0"  "0"             "0"                         
#> [2,] "0"  "((2 * r)) / 2" "0"                         
#> [3,] "0"  "0"             "((2 * r * sin(ph1)^2)) / 2"
#> 
#> , , 2
#> 
#>      [,1]            [,2]              
#> [1,] "0"             "( - (2 * r)) / 2"
#> [2,] "((2 * r)) / 2" "0"               
#> [3,] "0"             "0"               
#>      [,3]                                       
#> [1,] "0"                                        
#> [2,] "0"                                        
#> [3,] "((r^2 * (2 * (cos(ph1) * sin(ph1))))) / 2"
#> 
#> , , 3
#> 
#>      [,1]                         [,2]                                       
#> [1,] "0"                          "0"                                        
#> [2,] "0"                          "0"                                        
#> [3,] "((2 * r * sin(ph1)^2)) / 2" "((r^2 * (2 * (cos(ph1) * sin(ph1))))) / 2"
#>      [,3]                                          
#> [1,] "( - (2 * r * sin(ph1)^2)) / 2"               
#> [2,] "( - (r^2 * (2 * (cos(ph1) * sin(ph1))))) / 2"
#> [3,] "0"                                           
#>